Bending Space

As I hold a paper back book and bend it, I do not understand this idea of bending space. If I suppose the book and the air around it are really space, when I bend the book, that part of "space" goes into other parts of "space". Further, there must be space where the bending space once ways. What am I missing?

Staff: Mentor

Google for "intrinsic curvature" and "extrinsic curvature", and search the relativity forum here for those terms.

What you're doing is producing extrinsic curvature by deforming the two-dimensional surface of the book when you bend it through the third dimension. The space-time curvature that produces gravity in general relativity is the other kind - intrinsic curvature - and it doesn't need an extra dimension to bend through.

Staff: Mentor

Also be aware that what you're asking about is very, very non-intuitive. General relativity, which is the theory that describes spacetime and spacetime curvature, is based on very high-level mathematics about differential geometry. Differential geometry is about the study of non-flat geometries, like the surface of a saddle or a sphere. Non-flat geometry, also known as non-euclidean geometry, is NOT something people ever work with in school until well into college. Rules that hold in flat geometry (aka euclidean geometry) don't necessarily hold in non-euclidean geometry. For example, in basic geometry classes students learn that all three angles of a triangle add up to 180 degrees. If you study a triangle on a non-flat surface, you will find that this does NOT hold true.

If we extend this idea from a 2d surface to a 3d space then we get a very non-intuitive description of space but one that is perfectly logical in mathematical terms.

Staff: Mentor

Just about everyone is familiar with the nonflat geometry of the earth's curved surface. If two people standing at the south pole start walking in opposite directions, they'll meet face to face at the North pole. That's a curvature effect - if the surface of the earth were flat they'd never see each other again.

As I hold a paper back book and bend it, I do not understand this idea of bending space. If I suppose the book and the air around it are really space, when I bend the book, that part of "space" goes into other parts of "space". Further, there must be space where the bending space once ways. What am I missing?

To add, to what's already been said, don't feel badly about not "getting" this right off. We humans evolved in a world of Euclidean geometry(*) so there was zero survival value in understanding the non-Euclidean geometry of space-time.

* to be very technical, we evolved on a spherical surface (spherical geometry), but we didn't KNOW that for the first 99% of human evolution and so it had no relevance to human evolution